abstract: This dissertation reports three studies of the relationships between meanings teachers hold and meanings their students construct. The first paper reports meanings held by U.S. and Korean secondary mathematics teachers for teaching function notation. This study focuses on what teachers in U.S. and Korean are revealing their thinking from their written responses to the MMTsm (Mathematical Meanings for Teaching secondary mathematics) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. This paper then discusses how the MMTsm serves as a diagnostic instrument by sharing a teacher’s story. The data indicates that many teachers name rules instead of constructing representations of functions through function notation. The second paper reports the conveyance of meaning with eight Korean teachers who took the MMTsm. The data that I gathered was their responses to the MMTsm, what they said and did in the classroom lessons I observed, pre- and post-lesson interviews with them and their students. This paper focuses on the relationships between teachers’ mathematical meanings and their instructional actions as well as the relationships between teachers’ instructional actions and meanings that their students construct. The data suggests that holding productive meanings is a necessary condition to convey productive meanings to students, but not a sufficient condition. The third paper investigates the conveyance of meaning with one U.S. teacher. This study explores how a teacher’s image of student thinking influenced her instructional decisions and meanings she conveyed to students. I observed 15 lessons taught by a calculus teacher and interviewed the teacher and her students at multiple points. The results suggest that teachers must think about how students might understand their instructional actions in order to better convey what they intend to their students. The studies show a breakdown in the conveyance of meaning from teacher to student when the teacher has no image of how students might understand his or her statements and actions. This suggests that it is crucial to help teachers improve what they are capable of conveying to students and their images of what they hope to convey to future students. / Dissertation/Thesis / Doctoral Dissertation Mathematics Education 2019

Made available in DSpace on 2014-06-11T19:24:51Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-10-22Bitstream added on 2014-06-13T19:31:55Z : No. of bitstreams: 1 julio_rs_me_rcla.pdf: 746741 bytes, checksum: 5d17814f34e2626484e84678601aa91f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho fez-se uma leitura da produção de significados matemáticos e não-matemáticos para dimensão, baseado no Modelo dos Campos Semânticos (MCS), a qual constou de três episódios: (a) uma análise de frases do cotidiano que contêm a palavra dimensão, usando o MCS (Lins, 1997, 1999) e a noção de Jogos de Linguagem (Wittgenstein, 1985); (b) uma análise de como dimensão aparece na matemática do matemático, através de três definições matemáticas distintas para ela, e como estudantes de um curso de álgebra linear lidaram com essa noção; (c) um estudo histórico sobre aspectos de constituição de uma área específica da matemática buscando o que os historiadores falaram a respeito da noção de dimensão e das mudanças na produção de significados que aconteceram para essa noção. Esses episódios podem oferecer elementos para favorecer um diálogo com professores e futuros professores de matemática de tal forma que seja possível discutir diferentes modos de produção de significados para dimensão, particularmente entre discursos cotidianos e matemáticos, fazendo com que esses professores experienciem e discutam processos de produção de significados. / This study aimed at producing a reading of the production of mathematical and non-mathematical meanings for dimension, based on the Model of Semantic Fields (MSF), and consists of three episodes: (a) an analysis of everyday phrases that contain the word dimension, using both the MSF (Lins, 1997, 1999) and the notion of Language Games (Wittgenstein, 1985); (b) an analysis of how dimension appears within the mathematician's mathematics, through three definitions for it, and also re-examining how students of a linear algebra course dealt with it; and, (c) a historical study of the constitution of a specific area of mathematics, examining what historians say about the notion of dimension and the changes meanings went through. We think those episodes offer support for a dialogue with mathematics teachers (pre- and inservice) as to make possible to discuss those modes of meaning production, particularly the difference between everyday and mathematical discourses, providing an experience of better understanding meaning production processes.

Teachers' instructional decisions are important for students' mathematics learning as they determine the learning opportunities for all students. This study examines teachers' decisions about the activities and tasks they choose for students' mathematics learning, the ordering and connecting of mathematics topics, and the mathematics within curricula not to cover. These decisions are referred to as curricular decisions. I also identify teachers' mathematical schemes, referred to as mathematical meanings, in relation to geometric reflections and orientation of figures and examine teachers' reasoning with their mathematical meanings as they make these curricular decisions. Additionally, based on the results of this study I identify several productive and unproductive mathematical meanings in relation to geometric reflections and orientation of figures. Describing productive mathematical meanings as providing coherence to student mathematical understanding and preparing students for future mathematics learning (Thompson, 2016). These findings can be used to better understand why teachers make the curricular decisions they do as well as help teachers identify whether or not their mathematical meanings are productive in an effort to foster productive mathematical meanings for students.

abstract: This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

abstract: This dissertation reports three studies of students’ and teachers’ meanings for quotient, fraction, measure, rate, and rate of change functions. Each study investigated individual’s schemes (or meanings) for foundational mathematical ideas. Conceptual analysis of what constitutes strong meanings for fraction, measure, and rate of change is critical for each study. In particular, each study distinguishes additive and multiplicative meanings for fraction and rate of change. The first paper reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. Most teachers conveyed primarily additive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topics of targeted professional development. The second paper reports the quantitative part of a mixed method study of 153 calculus students at a large public university. The majority of calculus students not only have weak meanings for fraction, measure, and constant rates but that having weak meanings is predictive of lower scores on a test about rate of change functions. Regression is used to determine the variation in student success on questions about rate of change functions (derivatives) associated with variation in success on fraction, measure, rate, and covariation items. The third paper investigates the implications of two students’ fraction schemes for their understanding of rate of change functions. Students’ weak measurement schemes obstructed their ability to construct a rate of change function given the graph of an original function. The two students did not coordinate three levels of units, and struggled to relate partitioning and iterating in a way that would help them reason about fractions, rate of change, and rate of change functions. Taken as a whole the studies show that the majority of secondary teachers and calculus students studied have weak meanings for foundational ideas and that these weaknesses cause them problems in making sense of more applications of rate of change. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2016

Orientado: Romulo Campos Lins / Banca: Carlos Roberto Vianna / Banca: Rosa Lúcia Sverzut Baroni / Resumo: Neste trabalho fez-se uma leitura da produção de significados matemáticos e não-matemáticos para dimensão, baseado no Modelo dos Campos Semânticos (MCS), a qual constou de três episódios: (a) uma análise de frases do cotidiano que contêm a palavra dimensão, usando o MCS (Lins, 1997, 1999) e a noção de Jogos de Linguagem (Wittgenstein, 1985); (b) uma análise de como dimensão aparece na matemática do matemático, através de três definições matemáticas distintas para ela, e como estudantes de um curso de álgebra linear lidaram com essa noção; (c) um estudo histórico sobre aspectos de constituição de uma área específica da matemática buscando o que os historiadores falaram a respeito da noção de dimensão e das mudanças na produção de significados que aconteceram para essa noção. Esses episódios podem oferecer elementos para favorecer um diálogo com professores e futuros professores de matemática de tal forma que seja possível discutir diferentes modos de produção de significados para dimensão, particularmente entre discursos cotidianos e matemáticos, fazendo com que esses professores experienciem e discutam processos de produção de significados. / Abstract: This study aimed at producing a reading of the production of mathematical and non-mathematical meanings for dimension, based on the Model of Semantic Fields (MSF), and consists of three episodes: (a) an analysis of everyday phrases that contain the word dimension, using both the MSF (Lins, 1997, 1999) and the notion of Language Games (Wittgenstein, 1985); (b) an analysis of how dimension appears within the mathematician's mathematics, through three definitions for it, and also re-examining how students of a linear algebra course dealt with it; and, (c) a historical study of the constitution of a specific area of mathematics, examining what historians say about the notion of dimension and the changes meanings went through. We think those episodes offer support for a dialogue with mathematics teachers (pre- and inservice) as to make possible to discuss those modes of meaning production, particularly the difference between everyday and mathematical discourses, providing an experience of better understanding meaning production processes. / Mestre